Derivation and analysis on the analytical structure of interval type-2 fuzzy controller with two nonlinear fuzzy sets for each input variable<FootNote> Project supported by the Xinjiang Astronomical Observatory, China (No. 2014KL012), the Major State Basic Research Development Program of China (No. 2015CB857100), the National Natural Science Foundation of China (Nos. 51490660 and 51405362), and the Fundamental Research Funds for the Central Universities, China (No.SPSY021401) </FootNote>

Abstract

Type-2 fuzzy controllers have been mostly viewed as black-box function generators. Revealing the analytical structure of any type-2 fuzzy controller is important as it will deepen our understanding of how and why a type-2 fuzzy controller functions and lay a foundation for more rigorous system analysis and design. In this study, we derive and analyze the analytical structure of an interval type-2 fuzzy controller that uses the following identical elements: two nonlinear interval type-2 input fuzzy sets for each variable, four interval type-2 singleton output fuzzy sets, a Zadeh AND operator, and the Karnik-Mendel type reducer. Through dividing the input space of the interval type-2 fuzzy controller into 15 partitions, the input-output relationship for each local region is derived. Our derivation shows explicitly that the controller is approximately equivalent to a nonlinear proportional integral or proportional differential controller with variable gains. Furthermore, by comparing with the analytical structure of its type-1 counterpart, potential advantages of the interval type-2 fuzzy controller are analyzed. Finally, the reliability of the analysis results and the effectiveness of the interval type-2 fuzzy controller are verified by a simulation and an experiment.

Keywords

Interval type-2 fuzzy controller / Analytical structure / Karnik-Mendel type reducer

References

[1] Du, X.Y., Ying, H., 2007. Control performance comparison between a type-2 fuzzy controller and a comparable conventional Mamdani fuzzy controller. Annual Meeting of the North American Fuzzy Information Processing Society, p.100–105. http://dx.doi.org/10.1109/NAFIPS.2007.383819

[2] Du, X.Y., Ying, H., 2010. Derivation and analysis of the analytical structures of the interval type-2 fuzzy-PI and PD controllers. IEEE Trans. Fuzzy Syst., 18(4):802–814. http://dx.doi.org/10.1109/TFUZZ.2010.2049022

[3] Hagras, H., 2007. Type-2 FLCs: a new generation of fuzzy controllers. IEEE Comput. Intell. Mag., 2(1):30–43. http://dx.doi.org/10.1109/MCI.2007.357192

[4] Haj-Ali, A., Ying, H., 2004. Structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear PID control with variable gains. Automatica, 40(9):1551–1559. http://dx.doi.org/10.1016/j.automatica.2004.03.019

[5] Li, Q.C., Shen, D.Y., 2009. Brand-new PID fuzzy controller (fuzzy PI+fuzzy PD). Contr. Dec., 24(7):1037–1042 (in Chinese). http://dx.doi.org/10.13195/j.cd.2009.07.80.liqch.015

[6] Lin, F.J., 2015. Type-2 fuzzy logic control. IEEE Syst. Man Cybern. Mag., 1(1):47–48. http://dx.doi.org/10.1109/MSMC.2015.2395651

[7] Mendel, J.M., 2007. Type-2 fuzzy sets and systems: an overview. IEEE Comput. Intell. Mag., 2(1):20–29. http://dx.doi.org/10.1109/MCI.2007.380672

[8] Nie, M.W., Tan, W.W., 2012. Analytical structure and characteristics of symmetric Karnik-Mendel type-reduced interval type-2 fuzzy PI and PD controllers. IEEE Trans. Fuzzy Syst., 20(3):416–430. http://dx.doi.org/10.1109/TFUZZ.2011.2174061

[9] Wu, D.R., Tan, W.W., 2007. A simplified type-2 fuzzy controller for real-time control. ISA Trans., 45(4):503–516. http://dx.doi.org/10.1016/S0019-0578(07)60228-6

[10] Ying, H., Siler, W., Buckley, J.J., 1990. Fuzzy control theory: a nonlinear case. Automatica, 26(3):513–520. http://dx.doi.org/10.1016/0005-1098(90)90022-A

[11] Zhou, H.B., Ying, H., 2012. A technique for deriving analytical structure of a general class of interval type-2 TS fuzzy controllers. Annual Meeting of the North American Fuzzy Information Processing Society, p.1–6. http://dx.doi.org/10.1109/NAFIPS.2012.6290968

[12] Zhou, H.B., Ying, H., 2013. A method for deriving the analytical structure of a broad class of typical interval type-2 Mamdani fuzzy controllers. IEEE Trans. Fuzzy Syst., 21(3):447–458. http://dx.doi.org/10.1109/TFUZZ.2012.2226891

[13] Zhou, H.B., Ying, H., 2014. A method for deriving the analytical structure of the TS fuzzy controllers with two linear interval type-2 fuzzy sets for each input variable. IEEE Int. Conf. on Fuzzy Systems, p.612–618. http://dx.doi.org/10.1109/fuzz-ieee.2014.6891540

[14] Zhou, H.B., Ying, H., Duan, J.A., 2009. Adaptive control using interval type-2 fuzzy logic. IEEE Int. Conf. on Fuzzy Systems, p.836–841. http://dx.doi.org/10.1109/FUZZY.2009.5277302

RIGHTS & PERMISSIONS

2016 Zhejiang University and Springer-Verlag Berlin Heidelberg